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Publisher:

Dover Publications

Publication Date:

2010

Number of Pages:

384

Format:

Paperback

Edition:

2

Price:

16.95

ISBN:

9780486474175

Category:

Textbook

[Reviewed by , on ]

Allen Stenger

01/18/2011

The unusual and attractive feature of this book is that over half of the space is given to problem sequences. The book is organized as a collection of 33 short chapters, each one having a narrative section that deals with a single problem or concept, and a lengthy exercise section that further develops the ideas of the chapter. The present book is an unaltered reprint of the 1990 second edition.

The choice of topics and their treatment is very conventional, but the author has thought carefully about what the most important ideas are, and has highlighted those in the narratives. The chapters on Galois theory are especially good and really bring out the importance of field automorphisms, which I had never really appreciated before (having always gotten bogged down in the solvable-groups part of the theory).

Very Good Feature: topics are not introduced until they are needed, so you don’t have to wonder (for more than a few pages) why the author is telling you all this. For example, vector spaces are treated in the middle of field theory, because they are needed for field extensions.

On the down side, the exercises are not challenging; if a difficult problem needs to be solved, it is broken down into a series of simple steps. Most of the exercises are proofs rather than drill, and if you work through them all you will have a very thorough knowledge of the subject, but the process will not stretch you very much.

An even more stark book is Clark’s Elements of Abstract Algebra. This also strips the subject down to its most essential parts, but has only a few exercises. A more expansive survey book where everything is worked out, and that also has a good bit broader coverage, is Birkhoff & Mac Lane’s classic A Survey of Modern Algebra.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

**1 Why Abstract Algebra?**

History of Algebra

New Algebras

Algebraic Structures

Axioms and Axiomatic Algebra

Abstraction in Algebra

**2 Operations**

Operations on a Set

Properties of Operations

**3 The Definition of Groups**

Groups

Examples of Infinite and Finite Groups

Examples of Abelian and Nonabelian Groups

Group Tables

Theory of Coding: Maximum-Likelihood Decoding

**4 Elementary Properties of Groups**

Uniqueness of Identity and Inverses

Properties of Inverses

Direct Product of Groups

**5 Subgroups**

Definition of Subgroup

Generators and Defining Relations

Cayley Diagrams

Center of a Group

Group Codes; Hamming Code

**6 Functions**

Injective, Subjective, Bijective Function

Composite and Inverse of Functions

Finite-State Machines

Automata and Their Semigroups

**7 Groups of Permutations**

Symmetric Groups

Dihedral Groups

An Application of Groups to Anthropology

**8 Permutations of a Finite Set**

Decomposition of Permutations into Cycles

Transpositions

Even and Odd Permutations

Alternating Groups

**9 Isomorphism**

The Concept of Isomorphism in Mathematics

Isomorphic and Nonisomorphic Groups

Cayley’s Theorem

Group Automorphisms

**10 Order of Group Elements**

Powers/Multiples of Group Elements

Laws of Exponents

Properties of the Order of Group Elements

**11 Cyclic Groups**

Finite and Infinite Cyclic Groups

Isomorphism of Cyclic Groups

Subgroups of Cyclic Groups

**12 Partitions and Equivalence Relations**

**13 Counting Cosets**

Lagrange’s Theorem and Elementary Consequences

Survey of Groups of Order ≤10

Number of Conjugate Elements

Group Acting on a Set

**14 Homomorphisms**

Elementary Properties of Homomorphisms

Normal Subgroups

Kernel and Range

Inner Direct Products

Conjugate Subgroups

**15 Quotient Groups**

Quotient Group Construction

Examples and Applications

The Class Equation

Induction on the Order of a Group

**16 The Fundamental Homomorphism Theorem**

Fundamental Homomorphism Theorem and Some Consequences

The Isomorphism Theorems

The Correspondence Theorem

Cauchy’s Theorem

Sylow Subgroups

Sylow’s Theorem

Decomposition Theorem for Finite Abelian Groups

**17 Rings: Definitions and Elementary Properties**

Commutative Rings

Unity

Invertibles and Zero-Divisors

Integral Domain

Field

**18 Ideals and Homomorphisms**

**19 Quotient Rings**

Construction of Quotient Rings

Examples

Fundamental Homomorphism Theorem and Some Consequences

Properties of Prime and Maximal Ideas

**20 Integral Domains**

Characteristic of an Integral Domain

Properties of the Characteristic

Finite Fields

Construction of the Field of Quotients

**21 The Integers**

Ordered Integral Domains

Well-ordering

Characterization of Ζ Up to Isomorphism

Mathematical Induction

Division Algorithm

**22 Factoring into Primes**

Ideals of Ζ

Properties of the GCD

Relatively Prime Integers

Primes

Euclid’s Lemma

Unique Factorization

**23 Elements of Number Theory (Optional)**

Properties of Congruence

Theorems of Fermat and Euler

Solutions of Linear Congruences

Chinese Remainder Theorem

Wilson’s Theorem and Consequences

Quadratic Residues

The Legendre Symbol

Primitive Roots

**24 Rings of Polynomials**

Motivation and Definitions

Domain of Polynomials over a Field

Division Algorithm

Polynomials in Several Variables

Fields of Polynomial Quotients

**25 Factoring Polynomials**

Ideals of *F*[*x*]

Properties of the GCD

Irreducible Polynomials

Unique factorization

Euclidean Algorithm

**26 Substitution in Polynomials**

Roots and Factors

Polynomial Functions

Polynomials over Q

Eisenstein’s Irreducibility Criterion

Polynomials over the Reals

Polynomial Interpolation

**27 Extensions of Fields**

Algebraic and Transcendental Elements

The Minimum Polynomial

Basic Theorem on Field Extensions

**28 Vector Spaces**

Elementary Properties of Vector Spaces

Linear Independence

Basis

Dimension

Linear Transformations

**29 Degrees of Field Extensions**

Simple and Iterated Extensions

Degree of an Iterated Extension

Fields of Algebraic Elements

Algebraic Numbers

Algebraic Closure

**30 Ruler and Compass**

Constructible Points and Numbers

Impossible Constructions

Constructible Angles and Polygons

**31 Galois Theory: Preamble**

Multiple Roots

Root Field

Extension of a Field

Isomorphism

Roots of Unity

Separable Polynomials

Normal Extensions

**32 Galois Theory: The Heart of the Matter**

Field Automorphisms

The Galois Group

The Galois Correspondence

Fundamental Theorem of Galois Theory

Computing Galois Groups

**33 Solving Equations by Radicals**

Radical Extensions

Abelian Extensions

Solvable Groups

Insolvability of the Quintic

**Appendix A: Review of Set Theory**

**Appendix B: Review of the Integers**

**Appendix C: Review of Mathematical Induction**

**Answers to Selected Exercises**

**Index**

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